Introduction to Maxwell Equations

This section introduces the importance of Maxwell equations in understanding devices that utilize electricity and magnetism. The goal is to present the equations in a simple and understandable manner.

Understanding Electric and Magnetic Fields

• Electric and magnetic fields are fundamental concepts in Maxwell equations.

• Coulomb's law describes the electric force between charged spheres.

• The electric field E represents the force exerted by a source charge Q on a small charge q.

• The electric field is a vector with components Ex, Ey, and Ez indicating magnitude and direction.

Magnetic Field and Magnetic Force

• Moving charges in an external magnetic field experience a magnetic force.

• The magnetic force depends on the charge q, velocity v, and magnetic field B.

• The magnetic field B describes the strength of the external magnetic field.

Electric Field and Magnetic Field as Vector Fields

• Both electric field E and magnetic field B are vector fields.

• At each location (x, y, z), there is an associated electric and magnetic field vector indicating magnitude and direction.

Integral Form vs Differential Form of Maxwell Equations

This section discusses two different representations of Maxwell equations - integral form and differential form. Each form has its own advantages for calculating fields at single points or over entire regions in space.

Integral Form of Maxwell Equations

• Expresses Maxwell equations using integrals.

• Well-suited for computing fields over an entire region in space.

• Useful for solving symmetric problems.

Differential Form of Maxwell Equations

• Expresses Maxwell equations using derivatives.

• Useful for calculating fields at single points in space.

New Section

This section introduces the differential and integral forms of the electric field equations for charged objects. The differential form is more suitable for numerical calculations and deriving electromagnetic waves.

Divergence Integral Theorem

• The divergence integral theorem relates the flux of a vector field through a closed surface to the sources and sinks within a volume.

• The right-hand side of the equation represents a closed surface, denoted as A, enclosing a volume.

• F represents a vector field, such as an electric or magnetic field.

• da is an infinitesimal surface element with magnitude indicating area and direction orthogonal to the surface.

• The dot product between F and da represents the scalar product, which selects the component of F parallel to da.

• The integral sums up these scalar products over all locations on the surface A.

Surface Integral and Flux

• The right-hand side of the divergence integral theorem calculates the flux Phi of a vector field F through a closed surface A.

• If F is an electric field E, it's called electric flux. If F is a magnetic field B, it's called magnetic flux.

• V represents the volume enclosed by surface A.

• dv is an infinitesimal volume element within V.

• Nabla operator (∇) has three components corresponding to derivatives with respect to x, y, and z coordinates.

• Applying ∇ as a scalar product with F yields the divergence of F at each location (x,y,z), resulting in a scalar value.

Sources, Sinks, and Volume Integral

• Positive divergence indicates sources of vector field F at specific locations. Negative divergence indicates sinks.

• If enclosed by a surface, positive divergence leads to positive flux outwards while negative divergence results in negative flux inwards.

• Zero divergence implies no sources or sinks, with equal amounts of flux in and out.

• The left-hand side of the divergence integral theorem sums up the sources and sinks within a volume V using a volume integral.

New Section

This section focuses on understanding the divergence integral theorem and its role in comprehending Maxwell's equations.

Divergence Integral Theorem

• The divergence integral theorem relates the flux of a vector field through a closed surface to the sources and sinks within a volume.

• A closed surface A encloses any three-dimensional body, such as a cube or sphere.

• F represents an electric or magnetic field.

• da is an infinitesimal surface element with magnitude indicating area and direction orthogonal to the surface.

• The dot product between F and da represents the scalar product, which selects the component of F parallel to da.

• The integral sums up these scalar products over all locations on the surface A.

Understanding Flux

• The right-hand side of the divergence integral theorem calculates the flux Phi of a vector field F through a closed surface A.

• If F is an electric field E, it's called electric flux. If F is a magnetic field B, it's called magnetic flux.

• V represents the volume enclosed by surface A.

• dv is an infinitesimal volume element within V.

Nabla Operator (∇)

• ∇ (Nabla operator) has three components corresponding to derivatives with respect to x, y, and z coordinates.

• When applied as a scalar product with F, ∇ yields the divergence of F at each location (x,y,z), resulting in a scalar value.

Sources and Sinks

• Positive divergence indicates sources of vector field F at specific locations. Negative divergence indicates sinks.

• If enclosed by a surface, positive divergence leads to positive flux outwards while negative divergence results in negative flux inwards.

• Zero divergence implies no sources or sinks, with equal amounts of flux in and out.

Volume Integral

• The left-hand side of the divergence integral theorem sums up the sources and sinks within a volume V using a volume integral.

New Section

This section introduces the second important theorem, the curl integral theorem, which is necessary for understanding the Maxwell equations. It explains the concept of line integrals and how they measure the amount of a vector field that runs along a closed loop.

The Curl Integral Theorem

• The right-hand side of the equation represents a line integral along a closed loop L. The dl element is an infinitesimal line element of the loop.

• The scalar product between the vector field F and dl eliminates the orthogonal component and considers only the part parallel to dl.

• If F is an electric field, this line integral is referred to as electric voltage along L. If F is a magnetic field, it's called magnetic voltage along L.

• The left-hand side involves a surface A enclosed by the loop L. da is an infinitesimal piece of that surface.

• The cross product between Nabla operator and F gives the curl of F, which represents how much of F rotates around a point within A.

• According to the curl integral theorem, the total curl of F within A corresponds to the rotation of F along L.

New Section

This section introduces the first Maxwell equation in integral form. It explains how electric flux through a closed surface relates to enclosed charge.

First Maxwell Equation - Electric Flux

• The left-hand side represents a surface integral measuring electric flux through surface A using electric field E.

• The right-hand side is the total charge Q enclosed by A divided by the electric field constant.

• The first Maxwell equation states that electric flux through a closed surface corresponds to the enclosed electric charge.

New Section

This section introduces the second important theorem, the curl integral theorem, which is necessary for understanding the Maxwell equations. It explains the concept of line integrals and how they measure the amount of a vector field that runs along a closed loop.

The Curl Integral Theorem

• The right-hand side of the equation represents a line integral along a closed loop L. The dl element is an infinitesimal line element of the loop.

• The scalar product between the vector field F and dl eliminates the orthogonal component and considers only the part parallel to dl.

• If F is an electric field, this line integral is referred to as electric voltage along L. If F is a magnetic field, it's called magnetic voltage along L.

• The left-hand side involves a surface A enclosed by the loop L. da is an infinitesimal piece of that surface.

• The cross product between Nabla operator and F gives the curl of F, which represents how much of F rotates around a point within A.

• According to the curl integral theorem, the total curl of F within A corresponds to the rotation of F along L.

New Section

This section introduces the first Maxwell equation in integral form. It explains how electric flux through a closed surface relates to enclosed charge.

First Maxwell Equation - Electric Flux

• The left-hand side represents a surface integral measuring electric flux through surface A using electric field E.

• The right-hand side is the total charge Q enclosed by A divided by the electric field constant.

New Section

This section introduces the concept of charge density and explains how it relates to the volume integral of the charge. It also discusses the differential form of the first Maxwell equation.

Volume Integral and Charge Density

• The volume integral of the charge density rho over a volume V represents the charge enclosed in that volume.

• The right hand side of the Maxwell equation becomes a volume integral when considering charge density.

• The equation is satisfied for an arbitrarily chosen volume V if the integrands on both sides are equal, with the right integrand multiplied by 'one over epsilon_0'.

• This leads to the discovery of the differential form of the first Maxwell equation: 'the divergence of E is equal to rho over epsilon_0'.

New Section

This section focuses on understanding the divergence of electric field and its relationship with positive and negative charges.

Divergence of Electric Field

• The left hand side of the differential form shows that at a specific point in space, electric field divergence can be positive, negative, or zero.

• The sign of divergence determines whether there is a positive or negative charge at that point.

• Positive divergence indicates a positive charge density, while negative divergence indicates a negative charge density.

• Zero divergence suggests either no charge or an equal amount of positive and negative charges canceling each other out.

New Section

This section explores different scenarios based on electric field divergence values and their implications for charges in space.

Scenarios Based on Divergence

• A positive divergence implies a positive charge at that point, acting as a source for electric field.

• A negative divergence implies a negative charge at that point, acting as a sink for electric field.

• A zero divergence suggests no net charge or an equal amount of positive and negative charges canceling each other out.

• An ideal electric dipole could be located at a point with zero divergence.

New Section

This section discusses the second Maxwell equation in integral form, which involves magnetic fields and their flux through closed surfaces.

Second Maxwell Equation - Integral Form

• The second Maxwell equation involves a surface integral over A, but instead of an electric field, it focuses on a magnetic field B.

• According to the equation, the magnetic flux through a closed surface A is always zero.

• This implies that there are as many magnetic field vectors entering the surface as there are leaving it.

• The divergence integral theorem allows transforming the surface integral into a volume integral, introducing the concept of divergence of the magnetic field.

• For any volume V, the integral must be zero if the integrand is zero.

New Section

This section explains how the second Maxwell equation leads to its differential form and discusses implications for magnetic charges.

Divergence of Magnetic Field

• The differential form of the second Maxwell equation states that 'the divergence of B is equal to zero'.

• Zero divergence indicates that at every point in space (x, y, z), there are either no magnetic monopoles or an equal amount of positive and negative magnetic charges canceling each other out.

• Magnetic dipoles can exist with both north and south poles but not separated sources or sinks for the magnetic field.

• The absence of magnetic monopoles is a result derived from experimental observations.

New Section

This section emphasizes that according to the second Maxwell equation, there are no isolated sources or sinks for magnetic fields.

Absence of Magnetic Monopoles

• The absence of isolated sources or sinks for magnetic fields is a consequence of the second Maxwell equation in its differential form.

• Magnetic monopoles, which would represent single north or south poles without their corresponding counterparts, are not observed.

• The second Maxwell equation implies that only magnetic dipoles can exist, with both a north and a south pole.

New Section

This section highlights the symmetry and potential modifications of the second Maxwell equation if magnetic charges were discovered.

Symmetry and Potential Modifications

• If magnetic charges (magnetic monopoles) were found, the second Maxwell equation would need to be modified.

• The discovery of magnetic charges would make the Maxwell equations more symmetrical and potentially more beautiful.

• The absence of magnetic monopoles is an experimental result, but if they were detected, it would require reevaluating our understanding of electromagnetic phenomena.

New Section

This section introduces Faraday's law of induction as the third Maxwell equation in integral form.

Faraday's Law - Integral Form

• The third Maxwell equation is known as Faraday's law of induction.

• It involves a line integral of the electric field E over a closed line L that borders a surface A.

• This integral represents how much of the electric field rotates along the line L and corresponds to the electric voltage U along that line.

• On the right side, there is a surface integral of the magnetic field B over an arbitrary surface A.

New Section

This section explains how changes in magnetic flux through a surface relate to electric voltage along a closed line according to Faraday's law.

Relationship between Magnetic Flux and Electric Voltage

• The surface integral on the right side represents the magnetic flux Phi through surface A.

• Differentiating this flux with respect to time t gives the temporal change of the magnetic flux.

• The time derivative indicates how much the magnetic flux changes over time, which affects the rotating electric field.

• The minus sign accounts for the direction of rotation and ensures energy conservation according to Lenz's law.

New Section

This section discusses the relationship between changing magnetic flux and electric voltage along a closed line.

Relationship between Magnetic Flux Change and Electric Voltage

• A positive change in magnetic flux results in a negative electric voltage, while a negative change leads to a positive electric voltage.

• The behavior of electric voltage and magnetic flux change is opposite to each other.

• The minus sign in Faraday's law ensures energy conservation by preventing the amplification of the rotating electric field.

• This concept is known as Lenz's law, which states that the generated magnetic flux counteracts its cause.

New Section

This section summarizes Faraday's law and its implications for the relationship between rotating electric fields and time-varying magnetic fields.

Rotating Electric Field and Time-Varying Magnetic Field

• According to Faraday's law, a rotating electric field produces a time-varying magnetic field, and vice versa.

• Lenz's law explains that the generated magnetic flux opposes its cause to prevent energy creation from nothingness.

• In summary, Faraday's law states that changes in magnetic flux through a surface create an electric voltage along its edge.

New Section

This section highlights an important special case when there is no change in the magnetic field over time.

No Change in Magnetic Field

• If the magnetic field does not change over time, then the right side of Faraday's law is eliminated.

• In this case, it implies that the electric voltage along a closed line is always zero.

• Therefore, there is no rotating electric field as long as the magnetic field remains constant over time.

Understanding Electric Voltage

In this section, the concept of electric voltage and its relationship to energy transfer is explained. The curl integral theorem is introduced as a way to transform the integral form into the differential form.

Electric Voltage and Energy Transfer

• Electric voltage indicates the amount of energy gained or lost by a charge when it passes through a line.

• When the voltage is zero, there is no change in energy.

• The curl integral theorem connects a line integral with a surface integral, allowing for the transformation between integral forms.

The Third Maxwell Equation in Differential Form

This section focuses on deriving the differential form of the third Maxwell equation, which relates the curl of electric field E to the negative time derivative of magnetic field B. It highlights how changing magnetic fields generate rotating electric fields and vice versa.

Deriving the Differential Form

• The equation applies to any surface A, so both sides must be equal.

• The differential form states that a changing magnetic field B causes a rotating electric field E and vice versa while conserving energy.

Exploring the Fourth Maxwell Equation

Here, we delve into the fourth and final Maxwell equation, which involves line integrals of magnetic field B along closed lines. It introduces concepts such as magnetic voltage U, electric current I, and electric flux through a surface A.

Line Integral and Magnetic Voltage

• On the left side of the equation is a line integral of magnetic field B along a closed line L.

• This defines magnetic voltage U.

Electric Current and Electric Flux

• On the right side are constants Epsilon_0 (electric field constant) and Mu_0 (magnetic field constant).

• Electric current I is generated when electric charges flow along a conductor.

• The surface integral of the electric field represents the electric flux through surface A.

• The time derivative accounts for the temporal change of the electric flux.

Differential Form of the Fourth Maxwell Equation

This section focuses on deriving the differential form of the fourth Maxwell equation, which relates the curl of magnetic field B to current density j and changing electric fields.

Deriving the Differential Form

• Using the curl integral theorem, we transform the line integral into a surface integral, involving curl of magnetic field B.

• Current density j (current per area) is introduced to express current I as a surface integral.

• The scalar product ensures that only the parallel part of current density contributes to current through surface A.

Summary of Fourth Maxwell Equation in Differential Form

Here, we summarize all four Maxwell equations in their compact differential forms. These equations describe various relationships between electric and magnetic fields.

Compact Formulation

• First Maxwell equation: Divergence of E = charge density / epsilon_zero

• Second Maxwell equation: Divergence of B = 0 (no magnetic monopoles)

• Third Maxwell equation: Curl of E = -time derivative of B (changing magnetic field generates rotating electric field)

• Fourth Maxwell equation: Curl of B = mu_zero times I + mu_zero epsilon_zero times change in E over time (electric currents and changing electric fields generate magnetic fields)

Recap and Significance

In this final section, we reflect on what has been learned and highlight the significance and applications of Maxwells' equations in electrodynamics.

Recap and Applications

• Maxwells' equations form the foundation of electrodynamics.

• They encompass a wealth of knowledge and have numerous technical applications.

• The equations describe the generation and interaction of electric and magnetic fields.